We start with the same setup: assume that there are no two points of the same color which are one unit apart. If we start with any red point C, we can deduce that point D is red as well. We can use the same reasoning to show that there must be a red circle with center C and a radius of sqrt(3). There must be two points on this red circle which are exactly one unit apart. Contradiction, ergo our original assumption was incorrect. QED